I'm finding your use of "Black" and "White" as character names to be distracting. I realize that it's meant to be more straight-forward to use the chess sides as names, but it throws me off.

Give them names, give them genders (different genders is helpful for following things if it otherwise doesn't matter). Why? Because your reader cares about the emotional investment in the game and not the details, unless it's a reader who happens to be a chess expert. I know how to play chess in the casual way many do and I couldn't follow those details.

I suggest you intersperse the exchange with dialogue that describes the positions. "Rook to C1." If it's a formal game where the moves are called out, show it as actual quotes (from the player or a commentator). If it's an informal game, show the moves in italics as a description of the action. Then leave the narrative to describe the characters' emotions and strategy and so forth. This allows the reader to become invested in the moves and to understand them, even if they don't understand them.

(Note: I don't know chess notation and some moves may make no sense, just replace with accurate moves.)

After the last exchange, Hugo's position was a lot more comfortable,
and his opponent's more difficult.

White: Rook to C1.

This was the usual move in situations like this and Hugo expected Lida
to move her queen in response. Had he made the right move?

Black: Castle to A5.

Hugo bit his lip. He should have moved the rook to B1. He could do
it now, it set him up to revert to the Carlsbad structure if he had
to, but he'd lose a chance to move his knight into a more protective
position.

It depends on if you want to be precise or abstract.

If you want to be precise, proper notation (abc, 123) is the way to go, but this may lose your readers if they are unfamiliar with the notation. If you want to be abstract, describe it like the events aren't happening on a board, but as an actual battle that is happening around your players. This can still give the same feeling without being as constrained by the notation restrictions.

That said, it just really depends on what you want to go for.

The game should say something about the characters playing it.

What are you trying to convey through this scene? The decision-making process of one or both actors? The actual action of the game? How they perceive the struggle?

For example, if the main character is supposed to be seen as experienced, but perhaps not an expert in chess:

[Char one] didn't expect that move -- the King's Gambit. He had thought [char two] was the slow, strategic sort. This move, though, opened up risks for everyone. [Char one] glanced at the clock. There was no time to ponder -- he moved his queen's pawn to the center, as was his habit.

If a character is supposed to be seen as highly knowledgeable:

He opened with his favorite, the Queen's Gambit. If [char two] took the hanging pawn, he'd control the center of the board. From there, it would be easy. [Char two] declined, defending the center instead of taking the bait. "The Tarrasch defense," thought [Char one], "This will be a long game..."

In neither case am I describing the moves in detail.

If you want to describe an entire game through an extended scene, in detail, you should explore some completed games and base your scene on it. As long as you're basing your descriptions on a real game, you should have no problem sticking to what's possible.

Your story must be perfectly readable and understandable by people who do not play chess, do not know the rules, and only know through pop-culture osmosis that there are pieces called 'rook', 'knight', etc. Write with that in mind.

With that in mind, I probably wouldn't use chess notation at all. Somebody who has never played chess wouldn't be able to read it. Instead, I'd describe the situation, in broad strokes. A player might be forced to sacrifice a piece, or they might suddenly realise their careful plan has a fatal flaw, the opponent might find a way to escape a trap laid for them, or they might be playing an aggressive game, forcing the MC to do nothing but react. Those are all evocative descriptions that do not require the specifics of what's happening on the board, to be understood.

@Stackzebra mentions in a comment the chess game from Harry Potter and the Philosopher's Stone. It's a good example. The scene is engaging, it's fun, the reader experiences the tension of it, it works. Only trouble is, if you put the titbits of information about the game together, it turns out that not only is there no actual position in the author's mind that she's describing on the page, but she appears not to know the chess rules at all.

If I take one step forward

says Ron, the knight.

This example is great because the chess fails: the scene works despite the chess failure. Exemplifying what is important (the character dynamics, the sense of danger, etc.) and what isn't important (the chess).

For a different example, I would point you to the videogame Dragon Age Inquisition. While your player-character is walking around doing things, some of your companions banter in the background. And two of them start a chess game. Here's a link to the full dialogue, with added animation, and added overlay of the actual game being played. Incidentally, here the chess does work - they are playing the Immortal Game.

An example of dialogue from their game:

Solas: So, where were we? Ah, yes. Mage to C4.
Iron Bull: Little aggressive. Arishok to H4. Check.
Solas: Speaking of aggressive. I assume Arishok is your term for the Queen?

The two characters come from different cultures, their names for the pieces are different. It's a titbit of worldbuilding information that's interesting whether you're following the game or not. Also, note the commentary about a move being aggressive. Again, that clues in the non-chess-player audience. The game proceeds in the same way: it's all audio that you hear while playing, no board before you, so the dialogue must engage you in other ways, similar to how a novel would have to do it. In order to achieve that, every move is commented on, in a way that one doesn't need to understand chess in order to understand what's going on.

Iron Bull: You've got no Towers. You're down to a single Mage. Too bad you wasted time moving that Pawn to... to... You sneaky son of a bitch.

If your reader plays chess, you have no need to describe every piece. They will assume, unless otherwise stated, that most of the pieces are still in their original position.

Using the names of particular gambits and positions might be distracting. Remember, it is a game between two players and at certain levels, playing the player is important too.

The pawn exchange in the centre of the board opened him up to an
attack, his black knight sacrificed for position. Might have been a
blunder, his opponent seemed more confident. He saw a possible check
and decided to take it - moving his rook into position at KB8 - he
liked the old notation. Protected obliquely by his bishop, might just
pull a win out of this situation and flip it. Unless there was
something he didn’t see. Reluctantly, he removed his fingers from the
rook, committing to the move. Why was he smiling?

I'll answer this question from a chess perspective (I'm a National Master of chess). I don't think the description makes much sense. You indicated there is an isolated pawn in the center and that white is thinking of transitioning to a Carlsbad structure. The only way this could happen is if black has a pawn on d5, pawn on b7, and white can make an exchange on c6 where black could capture back with the pawn. In that case, it's hard to think of a reasonable situation where white would want his rook on b1 (unless I'm missing something). Your reader probably isn't going to notice but it bothers me when I see inaccurate descriptions of chess in media, though this isn't nearly as bad as most!

As a novelist myself, who writes fiction about the game of checkers, I find the critical question is this:

Who is your intended and expected audience?

Makes a lot of difference in how you handle it.

What is the purpose of this scene? I think that's the main question.

Is this a story for chess enthusiasts where the readers must follow every move and understand the strategic / tactical thoughts that went into it?

If the moves are integral to the story, I would say invest more words to describe the board and the scenario.

I am not an expert chess player by any means, but I play the game. And frankly, I can't follow the plot at all.

But if the chess game is not the point of the story, I would zoom out, gloss over the moves, and tell the actual story.

In all my experience as a reader, I have only seen ONE highly detailed description of an actual board game.... But that was because.. the game was integral to the story.

Let me give you an idea as a chessplayer myself. First off though, it was a good suggestion to use names instead of white and black. You can tell the reader who was white and who was black on the start of the game.
Do not describe a game. Describe the feelings. For example. The determination on the first moves on the opening part. The tension of a position in the middle game. The rush, the will to defeat one another, the whole battle of minds that is going on.
A surprising move that shocks the losing opponent. A suprprising comeback after a few moves. I mean, you can set up a chess game by describing feelings and not describing anything that happens on the chessboard.
Chess players will appreciate it, because they can relate to the agony, stress of a losing postion, the relief of a comeback, stuff like that. The rest of the readers don't have to know anything about chess to follow.

You either have to be precise, and include diagrams, or be "vaguely dramatic". There is no "middle ground" here.

Literature has many examples of chess matches and chess players. Other answers have already mentioned a number of them. What I think is common between them is that they are capturing the drama of the game, but not the game itself, not in enough detail that the reader is able to recreate it. You can mention the opening, and some individual moves, but the focus is on dramatic effect, not the moves themselves.

I know of only one example of "precise" chess fiction. Classic Russian Sci-Fi author Alexander Kazantsev had a book "Caissa's gift", in which fictional stories were revolving around chess games, complete with diagrams and move-by-move descriptions. I don't think this book was ever translated into English. The book, imho, was very interesting for both chess players and casual readers because one could either put the pieces on board and follow the story move by move, or ignore the exact moves and just follow the plot as it's written. It needs to be mentioned that Kazantsev was a master chess study composer.

Trying to provide some, but not all exact moves, and no diagrams I think would be a mistake. For a non-player, that would be simply a distraction. For a reader who is a chess player and would want to recreate the exact position, but that would prove to be very taxing, if possible at all.

Weave the position in the game into the larger theme of the novel. Don't use square locations like c1 and b1. Instead say something like "I moved my bishop one square and occupied the longest diagonal." or "My King couldn't move; I was checkmated." or "The rook made a lateral move; a blindspot I didn't notice."